The Theory of
Special Relativity has as its basic premise that light moves at a uniform
speed, c = 300,000 km/s, in all frames of reference. This results in
setting the speed of light as the absolute speed limit in the Universe and
also produced the famous relationship between mass and energy,
E = mc2.
The foundation of Einstein's General Theory is the
Equivalence Principle which states the equivalence between
inertial mass and gravitational mass.

Inertial Mass is the quantity that determines how difficult it is to alter the
motion of an object. It is the mass in Newton's Second Law:

F = ma

Gravitational mass is the mass which determines how strongly two objects
attract each other by gravity, e.g. the attraction of the earth:

It is the apparent equivalence of these two types of mass which results in the
uniformity of gravitational acceleration -- Galileo's result that all objects
fall at the same rate independent of mass:

Galileo and Newton accepted this as a happy coincidence, but Einstein turned
it into a fundamental principle. Another way of stating the equivalence
principle is that gravitational acceleration is indistinguishable from
other forms of acceleration. According to this view a student in a closed
room could not tell the difference between experiencing the gravitational pull
of the earth at the earth's surface and being in a rocketship in space
accelerating with a = 9.8 m/s2.

nor could students in a similar room distinguish between free-fall under
gravity and the weightlessness of space.

The second fundamental principle of General Relativity is that
the presence of matter curves space. In this view, gravity is not a
force, as described by Newton, but a curvature in the fabric of space, and
objects respond to gravity by following the curvature of space in the vicinity
of a massive object. The description of the curvature of space is the
mathematically complicated part of general relativity involving "metrics",
which describe the way that matter curves space, and tensor calculus.

The Curvature of Space caused by a Massive Object.

The figure above represents a two-dimensional slice through three-dimensional
space showing the curvature of space produced by a spherical object, perhaps
the sun. Einstein's view is that the planets follow the curvature of space
around the sun (and produce a tiny amount of curvature themselves).

Here are two fine astronomy course pages on General Relativity from Dr.
Terry Herter at
Cornell, from whom I stole the above images, and astronomers at the
University of Tennessee.

Deflection of Light by Gravity: A direct consequence of
the equivalence principle is that light should be deflected or bent by
gravity. Einstein twice calculated the amount that light would be
deflected passing by the sun, the largest "nearby" mass. His first
calculation used only the Equivalence Principle and the equivalent
mass-energy of a visible photon. In his second calculation, published in
1916, he included the space-time metric, which describes the curvature of
space and time caused by gravity and got an answer twice as large as his first
calculation. The second calculation predicts that light from a distant star
passing by the limb of the sun would be deflected by 1.75 arcseconds (less
than 1/2000th of a degree).

The first opportunity to test Einstein's calculation came with the Solar
Eclipse of 1919. British Astrophysicist Sir Arthur Eddington mounted
a pair of expeditions to West Africa and Brazil to observe the shift in
position of the Hyades cluster stars behind the occulted sun. Eddington's
measurements, though not perfectly precise clearly showed a deflection and
favored the larger value. The result made Einstein world-famous. The test
can now be made with greater precision. Every year the radio source 3C279
is occulted by the sun. Because the sun is only a modest radio-emitter,
Radio Astronomers do not need to wait for an eclipse. Radio interferometry
of 3C279 as it passes behind the sun has confirmed Einstein's calculation
to better than 1%.

An exciting and only very recently verified prediction of the bending of light
by gravity is the existence of
gravitational lenses; an optical lens focuses light be refraction,
bending of light due to the change of the speed of light as it passes through
a refractive medium. Because gravity can bend light, massive objects can act
as lenses, focusing and amplifying images of distant objects. Gravitational
lenses have rather different properties than "normal" lenses producing
multiple images such as the
Einstein Cross,
a case of a distant quasar imaged by a galaxy between us and the quasar,
discovered by J. Huchra & colleagues, shown to the left. If the alignment
between us, the lensing galaxy, and the distant object, an
Einstein Ring is
produced. Distant
galaxy clusters may also act as gravitational lenses. Astronomers are
beginning to make use of the gravitational lensing phenomenon to study very
distant galaxies and quasars. More about this in Lecture #17.

Gravitational Redshift: Light loses energy escaping from a
gravitational field. Because the energy of light is proportional to its
frequency, a shift toward lower energy represents a shift to lower frequency
and longer wavelength or a shift toward the red for visible light. This
gravitational redshift was first observed for the absorption features from
White Dwarf stars, whose light is shifted by about 1Å. It
was experimentally verified on earth using gamma-rays travelling from the
basement to the top of the Jefferson Tower Physics Laboratories at Harvard.

Gravitational Time Dilation: Time slows down in a strong
gravitational field. Newton's concept of time was an absolute quantity
flowing uniformly through the Universe. Einstein showed that the measurement
of time is relative, depending on the reference frame of the person
who is making the measurement. The Special Theory demonstrated that
timekeepers in motion with respect for each other will measure different
times for events in each others reference frames: a timekeeper "at rest"
will find that an event occurring in a rapidly moving reference frame will
take a longer time than a timekeeper moving along with the event. A famous
example of Special Relativistic Time Dilation is the
Twin Paradox:

Twins Bill and Jill, born within minutes of each other, take differing career
paths. Jill becomes an astronaut and Bill becomes a ground-based astronomer.
On their 21st birthday Jill sets out on a space mission to Aldebaran, 32 light
years away. Travelling at 99.5% of the speed of light, Jill measures a time of
3.2 years for her trip to Aldebaran and another 3.2 years for her return.
(Incideltally, while she is travelling near the speed of light she also sees
the distance to Aldebaran contracted to a mere 3.2 light years.)
Bill finds that it takes her 32 years and 2 months for each leg. Upon Jill's
return, she is 27 while her sibling is 85! Bizarre as these effects appear to
us slow moving mortals, relativistic time dilation has been
repeatedly confirmed in high energy particle accelerators, where particles
travel near the speed of light, and by atomic clock on supersonic aircraft.

A similar process occurs in the presence of strong gravity; a timekeeper in
a strong gravitational field will measure a slower time than one in the
absence of gravity. It is not just clocks, by the way, all physical processes:
clocks ticking (however they measure their ticks), hearts beating, aging, etc.,
must slow down, but the only one who notices is the distant timekeeper.
Everything seems "normal" to the person measuring the duration of events in
his own frame of reference. Light waves travelling past the sun are slowed
down by this time dilation by a small but measurable amount. In 197X the
Viking Mars Lander performed the initial confirming experiment of
gravitational time dilation by relaying radio signals back to earth
from the Martian surface on the other side of the solar system. Although
the effects of the intervening solar wind complicate the experiment, NASA
scientists demonstrated clearly that the radio signals took longer on their
round trip by just the amount predicted by the predicted slowing of time.

Precession of Mercury's Orbit: Kepler's first law states
that planets travel around the sun in elliptical orbits and Newton verified
this as a consequence of his law of gravity. Even in Newtonian physics this
law is not obeyed precisely, because the gravitational pull of the other
planets perturbs the orbit in a small but detectable way; It was by these
small perturbations that the planets Neptune and Pluto were discovered. It
was known well before Einstein that these effects cause the axis of the orbit
of Nercury to rotate slowly, so that its point of closest approach to the sun,
the perihelion, moves or precesses from orbit to orbit, as shown
in the figure to the right from NCSA's Relativity site. The motion is very
small - it takes a quarter of a million years for Mercury's perihelion to
complete a full circle around the sun. However, the effects of the known
planets cannot account for all of Mercury's precession. The only plausible
explanation in Newton's gravity is another planet, even closer to the sun
than Mercury. This planet, prematurely named Vulcan, was never found despite
exhaustive searches and the extra precession - about 1/100th of a degree per
century - remained a mystery. Einstein applied his General Theory to the
motion of Mercury and found that the somewhat higher gravitational pull
as the planet approaches the sun in General Relativity causes Mercury to
move a bit further around the sun each time it passes. His calculation
found exactly the observed extra precession. Because the precession of
Mercury's orbit is a direct result of the full General Theory, not just the
Equivalence Principle, Einstein viewed it as the most critical test of his
theory.

Gravitational Radiation: Just as accelerating
charged particles produces disturbances in the electromagnetic field -
electromagnetic radiation (light) - accelerating masses produce
disturbances in the gravitational field - gravitational radiation. In
General Relativity gravity is viewed as a curvature of Spacetime so
Gravitational
Waves are ripples in the fabric of space and time itself. A
gravitational wave alternately stretches and shrinks space as it passes
through, but on a very small scale (as little as a factor of 10-21
even for a very strong source).

Predicted sources of strong gravitational waves in the Galaxy are
supernova explosions,
collapsing stellar cores as they form neutron stars or black holes,
compact binary star systems,
collisions of neutron stars & black holes, or possibly material
falling into the blavk hole which may reside in the Galactic Center.
Gravitational waves have not yet been detected directly, but we believe that
they have been detected indirectly by radio astronomers in the
binary
pulsar system 1913+16. As the pulsar is accelerated around its companion,
orbiting every 8 hours in this compact system, General Relativity predicts
that gravitational waves should be produced. Although these waves are far too
faint to be detected directly, the binary pulsar system is losing energy
through this radiation, and the pulsar/neutron star and its companion
are predicted to be slowly spiralling together. The rapid radio pulses
permit precise timing of the pulsar orbit by doppler shifts of the pulse
period as the pulsar moves toward or away from us. Since the discovery of
the binary pulsar in 1974, timing of the pulsar has shown that the stars are
indeed spiralling together just as predicted. In 300 million years the
stars will coalesce - that should produce gravitational radiation that can be
easily detected!

Laser interferometer observatories are being constructed that will be
able to detect gravitational waves from possible sources potentially as far
away as 100 million light years. A collaboration between Caltech and MIT is
building LIGO, the Laser
Interferometric Gravitational-Wave Observatory, to begin observations in 2000
and there are at least five other projects among European, Australian,
Japanese, and Space Physicist groups constructing other Gravitational-Wave
Observatories. Watch this space for News!

All of this amounts to pretty spectacular confirmation of General Relativity
Theory.

So, Einstein was right and Newton was wrong!

Well, that's not really the way science works and besides, Newton was not
completely wrong and Einstein is not completely right. Newton's Theory is
perfectly fine for most calculations of gravity where the field is not too
strong. NASA scientists do not use General Relativity to calculate the
paths of spacecraft that are sent to explore the solar system (not because it
would be too complicated or difficult, but because it would be a waste of time
- Newton was right as far as most things in the Solar System are concerned).
Furthermore, General Relativity fails on very small scales when quantum
mechanical effects become important. Theoretical Physicists are working
very hard on theories of "Quantum Gravity", but so far no one has succeeded
in improving Einstein's Theory; no doubt the next better theory of gravity
will still be an approximation of the "Truth". Science works by:

developing theories or hypotheses,

testing them repeatedly by experiment and observation,

using them where they are shown to be applicable, and

revising & improving them when they are shown to disagree with
experiment.

will have an escape speed equal to the speed of light. We call
such an object a
Black Hole. (Note that for the sun to be a black hole it would have to be compressed by a quarter of a million times down to a
radius less than 3km.) A black hole is an object so compact that nothing
can escape its gravity, not even light. Mathematically, a black hole is
an object of zero size and infinite density (but finite mass) - a
singularity. Schwarzschild's calculation shows that the gravitational
radius, also called the Schwarzschild radius or event
horizon, provides an effective size for a black hole because nothing
can escape from inside the gravitational radius and there can be no
communication from objects inside Rgrav and the outside world.

Curved Spacetime around a Black Hole.
Inside the horizon or gravitational radius space
is so strongly curved that nothing can escape.

First, perhaps we should dispel a prime misapprehension about black holes:
Black holes are not gigantic vacuum cleaners sucking everything in the
Universe into their darkness. And you would have to be pretty foolish to
get caught in the strong gravity of a black hole; hopefully our interstellar
astronauts will get better training than the hapless space explorers in so
many bad sci-fi stories. This is because black holes have finite
mass and because everything in the Universe is so far apart. Black holes are
produced by massive stars as a natural part of the stellar evolutionary
process. A black hole from a collapsed 10M stellar core will have a mass
of 10 solar masses. It will produce gravitational effects on neighboring
stars just like a normal 10M star would. You need to get close
to black hole (i.e. near the gravitational radius) for its strong
gravity to "suck you in" or for General Relativistic effects to be important.

Similarly, if you were on a planet orbiting a star which became a black hole,
you would not be sucked in by the Black Hole's gravity. If the star loses
no mass, you would feel no change in the gravity and would continue to stay in
the same orbit. (Lots of other bad things would happen, particularly if the
star goes through a supernova explosion. In that case, cosmic rays &
gamma rays would extinguish life on the planet and the mass lost in the
explosion would decrease the gravitational pull of the remnant
causing your planet to fly off into space.)

Should you be unfortunate enough (and foolish enough) to be sucked into a
black hole your demise would be nearly instantaneous in your reference
frame. You would first be pulled apart by intense tidal forces because
the force on your feet would be much stronger than that on your head. Then
you would be crushed into infinite density as you become part of the
singularity at the center of the black hole.

Things would appear very different to a distant observer due to the General
Relativistic effects described above. Time, as measured far away, would
appear to get slower and slower as you approach the gravitational radius -
the distant observer would be denied the thrill of actually seeing you
disappear into the black hole because time "stops" at the gravitational
radius. Any signals that you send would be redshifted by increasing amounts
as you near the black hole.

We believe that we have found black holes in our galaxy in the form of
X-Ray Binary Stars. In these star systems material may be transferred
from a main sequence or red giant companion onto the black hole. (Remember
that massive stars live fast & die young.) When a binary star system is
formed, the more massive star will complete its life cycle first, becoming a
black hole (or perhaps a neutron star). When the lower mass companion begins
to expand, evolving toward the red giant phase, material may be pulled toward
the black hole. Because of the angular momentum from the stars mutual orbits,
the material cannot fall directly down the black hole, but spirals inward
forming an accretion disk. The release of gravitational energy as
material spirals into the black hole heats the accretion disk to millions of
degrees so that it emits x-rays.

Artists Conception of the Black Hole Binary Star System, Cygnus X-1.
Material is pulled from the Companion into an Accretion Disk (shown in red)
which is heated to millions of degrees as material spirals into the Black Hole.

Neutron stars in binary star systems may also be x-ray binaries. Material
falling from a companion onto a compact neutron star may release just about as
much gravitational energy as material falling into a black hole. Neutron stars
will probably be pulsars in x-rays just like in the radio. Here is a JAVA
x-ray pulsar
animation courtesy of the Chandra X-Ray Observatory.

The best known black hole candidate is
Cygnus X-1,
an x-ray binary in Cygnus and one of the brightest x-ray sources in the sky.
In 1972 Cygnus X-1 was identified with a 9th magnitude O supergiant,
catalogued as HDE226868. HDE226868 is orbiting an unseen companion which
orbital analysis indicates has a mass of about
20M,
far too massive to be a neutron star or white dwarf. Cygnus X-1 also has
unusual
x-ray properties
which lend support to the idea that this must be a black hole.

Stellar black holes have masses in the range of a few times the mass of the
sun, up to a few tens of solar masses, but other processes may produce very
massive black holes. There is increasing evidence that there may be a million
solar mass black hole in the center of our Milky Way
galaxy, and black holes with masses up to a billion times the sun's mass in
the cores of other galaxies. Many astronomers also believe that black holes
power quasars and other active galaxies.

Black Hole Links & References

The best book about black holes is Kip Thorne's "Black Holes &
Time Warps: Einstein's Outrageous Legacy" (W.W. Norton, 1994). This book
is challenging, but worth the effort.